Conjecture:

“What restrictions can be placed on the maximum and minimum number of visits per vertices?”

Rules:

Each vertices can visit any other vertices a maximum number of one times, there can be only one connecting line between any two vertices. All vertices must be connected by one continues non-repetitive line.

Observations:

There exist three types of vertices. A starting, central, and end vertices. No vertices may be more than one of these three types..

Restrictions:

Starting or ending vertices maximum number of connecting lines is equal to the total number of vertices minus one. The minimum number of connecting lines, or visits, is equal to one.

A non-starting or non-ending vertices has the maximum number of connecting lines equal to the total number of vertices minus one. The minimum number of connecting lines is equal to two, because central vertices must have at least two connecting vertices. If there are less than three vertices, then DNE.

Thought Process:

The first step of my thought process was to decode the conjecture to figure out exactly what was being asked or looked for. The issue was that I did not like the phrasing of our question. It would have been simpler and more direct to ask, “How many lines can be connected to each vertices if we have the following restrictions…?” Once I reworded the phrase I could start making progress. My partner, Neil, had already come up with the notion that there were three different types of vertices so I reasoned that I would need to show the maximum and minimum number of connecting lines that each vertices had. After drawing a handful of vertices and connecting lines I came to the realization that starting and ending points could be treated in the same fashion. That is, they both had identical restrictions. At first, I was going to lump in the central vertices with these two, then I realized that vertices group had to be treated different. The minimum number of trips seemed obvious to me, but I didn’t think of the minimum number of vertices restriction until later. I was quite pleased with myself for figuring that little tidbit out. It just occurred to me to define why central vertices have a minimum of two visits.

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